Two-Person Games

Game theory was developed, starting in the 1940’s, as amodel of situations of conflict. Such situations andinteractions will be called games and they have participantswho are called players. We will focus on games withexactly two players. These two players compete for a payoffthat one player pays to the other. These games are calledzero-sum games because one player’s loss is the otherplayer’s gain, and the payoff to both players for any givenscenario adds to zero.

A coin-matching gameRoger and Colleen play a game. Each one has a coin. Theywill both show a side of their coin simultaneously. If bothshow heads, no money will be exchanged. If Roger showsheads and Colleen shows tails then Colleen will give Roger$1. If Roger shows tails and Colleen shows heads, thenRoger will pay Colleen $1. If both show tails, then Colleenwill give Roger $2.

This is a two-person game, with players Roger and Colleen.It is also a zero-sum game. This means that Roger’s gain isColleen’s loss (and vice-versa).

Like chess, this is a game where both players have fullknowledge of what is happening. Unlike chess, it is a gamewhere both players play simultaneously, as opposed toalternately.

A coin-matching gameRoger and Colleen play a game. Each one has a coin. Theywill both show a side of their coin simultaneously. If bothshow heads, no money will be exchanged. If Roger showsheads and Colleen shows tails then Colleen will give Roger$1. If Roger shows tails and Colleen shows heads, thenRoger will pay Colleen $1. If both show tails, then Colleenwill give Roger $2.

This is a two-person game, with players Roger and Colleen.It is also a zero-sum game. This means that Roger’s gain isColleen’s loss (and vice-versa).

Like chess, this is a game where both players have fullknowledge of what is happening. Unlike chess, it is a gamewhere both players play simultaneously, as opposed toalternately.

A coin-matching gameRoger and Colleen play a game. Each one has a coin. Theywill both show a side of their coin simultaneously. If bothshow heads, no money will be exchanged. If Roger showsheads and Colleen shows tails then Colleen will give Roger$1. If Roger shows tails and Colleen shows heads, thenRoger will pay Colleen $1. If both show tails, then Colleenwill give Roger $2.

This is a two-person game, with players Roger and Colleen.It is also a zero-sum game. This means that Roger’s gain isColleen’s loss (and vice-versa).

Like chess, this is a game where both players have fullknowledge of what is happening. Unlike chess, it is a gamewhere both players play simultaneously, as opposed toalternately.

A coin-matching game

We can use a 2 × 2 array to show all four situations thatcan arise in a single play of this game, and the results ofeach situation, as follows:

Colleen

Roger

Heads Tails

HeadsRoger pays/gets $0

Colleen pays/gets $0Roger gets $1

Colleen pays $1

TailsRoger pays $1Colleen gets $1

Roger gets $2Colleen pays $2

The amount won by either player in any given situation iscalled the pay-off for that player.

A coin-matching game

We can use a 2 × 2 array to show all four situations thatcan arise in a single play of this game, and the results ofeach situation, as follows:

Colleen

Roger

Heads Tails

HeadsRoger pays/gets $0

Colleen pays/gets $0Roger gets $1

Colleen pays $1

TailsRoger pays $1Colleen gets $1

Roger gets $2Colleen pays $2

The amount won by either player in any given situation iscalled the pay-off for that player.

A coin-matching gameBecause this is a zero-sum game, we can deduce the pay-offfor one player from that of the other, and so we can deduceall information about the game from the pay-off matrixshown below:

Colleen

Rog.

H TH 0 1T −1 2

The pay-off matrix for a game shows only the pay-off forthe row player — this is a universal convention. A positivepay-off for the row player indicates that he gains thatamount (and the column player losses that amount). Anegative pay-off for the row player indicates that he losesthat amount (and the column player gains that amount).

A coin-matching gameBecause this is a zero-sum game, we can deduce the pay-offfor one player from that of the other, and so we can deduceall information about the game from the pay-off matrixshown below:

Colleen

Rog.

H TH 0 1T −1 2

The pay-off matrix for a game shows only the pay-off forthe row player — this is a universal convention. A positivepay-off for the row player indicates that he gains thatamount (and the column player losses that amount). Anegative pay-off for the row player indicates that he losesthat amount (and the column player gains that amount).

A coin-matching gameColleen

Rog.

H TH 0 1T −1 2

A player’s plan of action against the opponent is called astrategy.

In the present example, each player has two possiblestrategies; H and T. If, for example, Roger employs thestrategy H and Colleen employs T, then the payoff is 1 —Roger receives $1, Colleen loses $1.

Our goal is to determine each player’s best strategy,assuming both players want to maximize their pay-off. Ourconclusions will make most sense when we consider playerswho are repeatedly playing the same game.

A coin-matching gameColleen

Rog.

H TH 0 1T −1 2

A player’s plan of action against the opponent is called astrategy.

In the present example, each player has two possiblestrategies; H and T. If, for example, Roger employs thestrategy H and Colleen employs T, then the payoff is 1 —Roger receives $1, Colleen loses $1.

Our goal is to determine each player’s best strategy,assuming both players want to maximize their pay-off. Ourconclusions will make most sense when we consider playerswho are repeatedly playing the same game.

Summary of assumptions in generalI We will limit our attention to two-person zero-sum

games.

I We assume that each player is striving to maximizetheir pay-off.

I Each player will have several options or strategies that(s)he can exercise.

I Each time the game is played, each player selects oneoption/strategy.

I The players decide on their strategies simultaneouslyand independently.

I Each player has full knowledge of the strategiesavailable to himself and his opponent and the pay-offsassociated to each possible scenario. (However neitherplayer knows which strategy their opponent willchoose.)

The language of the pay-off matrix

For a two-player, zero-sum game, we will call the twoplayers R (for row) and C (for column).

For each such game, we can represent all of the informationabout the game in a matrix, called the pay-off matrix.

It is an array with a list of R’s strategies as labels for therows and a list of C’s strategies as labels for the columns.The entries in the pay-off matrix are what R gains for eachcombination of strategies. If this is a negative number thanit represents a loss for R.

N.B. — the pay-off matrix will always be presented fromR’s point of view. This is a completely arbitrary choice.

The language of the pay-off matrix

For a two-player, zero-sum game, we will call the twoplayers R (for row) and C (for column).

For each such game, we can represent all of the informationabout the game in a matrix, called the pay-off matrix.

It is an array with a list of R’s strategies as labels for therows and a list of C’s strategies as labels for the columns.The entries in the pay-off matrix are what R gains for eachcombination of strategies. If this is a negative number thanit represents a loss for R.

N.B. — the pay-off matrix will always be presented fromR’s point of view. This is a completely arbitrary choice.

The language of the pay-off matrix

For a two-player, zero-sum game, we will call the twoplayers R (for row) and C (for column).

For each such game, we can represent all of the informationabout the game in a matrix, called the pay-off matrix.

It is an array with a list of R’s strategies as labels for therows and a list of C’s strategies as labels for the columns.The entries in the pay-off matrix are what R gains for eachcombination of strategies. If this is a negative number thanit represents a loss for R.

N.B. — the pay-off matrix will always be presented fromR’s point of view. This is a completely arbitrary choice.

The language of the pay-off matrix

For a two-player, zero-sum game, we will call the twoplayers R (for row) and C (for column).

For each such game, we can represent all of the informationabout the game in a matrix, called the pay-off matrix.

It is an array with a list of R’s strategies as labels for therows and a list of C’s strategies as labels for the columns.The entries in the pay-off matrix are what R gains for eachcombination of strategies. If this is a negative number thanit represents a loss for R.

N.B. — the pay-off matrix will always be presented fromR’s point of view. This is a completely arbitrary choice.

Two Finger MorraRuth and Charlie play a game. At each play, Ruth and Charliesimultaneously extend either one or two fingers and call out anumber. The player whose call equals the total number ofextended fingers wins that many pennies from the opponent. Inthe event that neither player’s call matches the total, no moneychanges hands.

Here’s the pay-off matrix for this game (here strategy (1, 2)means that the player holds up one finger and shouts “2”).

Charlie

Ruth

(1, 2) (1, 3) (2, 3) (2, 4)

(1, 2) 0 2 −3 0(1, 3) −2 0 0 3(2, 3) 3 0 0 −4(2, 4) 0 −3 4 0

Two Finger MorraRuth and Charlie play a game. At each play, Ruth and Charliesimultaneously extend either one or two fingers and call out anumber. The player whose call equals the total number ofextended fingers wins that many pennies from the opponent. Inthe event that neither player’s call matches the total, no moneychanges hands.

Here’s the pay-off matrix for this game (here strategy (1, 2)means that the player holds up one finger and shouts “2”).

Charlie

Ruth

(1, 2) (1, 3) (2, 3) (2, 4)

(1, 2) 0 2 −3 0(1, 3) −2 0 0 3(2, 3) 3 0 0 −4(2, 4) 0 −3 4 0

Two Finger Morra

Charlie

Ruth

(1, 2) (1, 3) (2, 3) (2, 4)(1, 2) 0 2 −3 0(1, 3) −2 0 0 3(2, 3) 3 0 0 −4(2, 4) 0 −3 4 0

(b) What is the payoff for Ruth if Ruth shows two fingersand calls out 4 and Charlie shows 1 finger and calls out 3?What is the payoff for Charlie in this situation?

(2, 4) for Ruth is row 4 and (1, 3) for Charlie is column 2.Hence −3 is Ruth’s payoff so she gives Charlie 3 cents orCharlie’s payoff is 3.

Two Finger Morra

Charlie

Ruth

(1, 2) (1, 3) (2, 3) (2, 4)(1, 2) 0 2 −3 0(1, 3) −2 0 0 3(2, 3) 3 0 0 −4(2, 4) 0 −3 4 0

(b) What is the payoff for Ruth if Ruth shows two fingersand calls out 4 and Charlie shows 1 finger and calls out 3?What is the payoff for Charlie in this situation?

(2, 4) for Ruth is row 4 and (1, 3) for Charlie is column 2.Hence −3 is Ruth’s payoff so she gives Charlie 3 cents orCharlie’s payoff is 3.

Rock-Paper-Scissors

The players face each other and simultaneously displaytheir hands in one of the three following shapes: a fistdenoting a rock (R), the forefinger and middle fingerextended and spread denoting scissors (S), or a downwardfacing palm denoting a sheet of paper (P). Rock beats(smashes) Scissors, Scissors beats (cuts) Paper, and Paperbeats (covers) Rock. The winner collects a penny from theopponent and no money changes hands in the case of a tie.

Here’s the pay-off matrix:R P S

R 0 −1 1P 1 0 −1S −1 1 0

Rock-Paper-Scissors

The players face each other and simultaneously displaytheir hands in one of the three following shapes: a fistdenoting a rock (R), the forefinger and middle fingerextended and spread denoting scissors (S), or a downwardfacing palm denoting a sheet of paper (P). Rock beats(smashes) Scissors, Scissors beats (cuts) Paper, and Paperbeats (covers) Rock. The winner collects a penny from theopponent and no money changes hands in the case of a tie.

Here’s the pay-off matrix:R P S

R 0 −1 1P 1 0 −1S −1 1 0

Football: Run or Pass?The Ravens are playing the Cowboys, and the Ravens are onoffense. The Ravens select a play and the Cowboys line up todefend. The choices are made simultaneously, with each teamhaving no knowledge of the other team’s choice.

Here’s a simple model: the offense chooses either to run or topass, and the defense chooses to defend a run or a defend a pass.

Based on historical data,

I if the Ravens run, and the Cowboys defend the run, theRavens lose 5 yards on average;

I if the Ravens run, and the Cowboys defend the pass, theRavens gain 5 yards on average;

I if the Ravens pass, and the Cowboys defend the pass, theRavens gain 0 yards on average;

I if the Ravens pass, and the Cowboys defend the run, theRavens gain 15 yards on average.

Football: Run or Pass?The Ravens are playing the Cowboys, and the Ravens are onoffense. The Ravens select a play and the Cowboys line up todefend. The choices are made simultaneously, with each teamhaving no knowledge of the other team’s choice.

Here’s a simple model: the offense chooses either to run or topass, and the defense chooses to defend a run or a defend a pass.

Based on historical data,

I if the Ravens run, and the Cowboys defend the run, theRavens lose 5 yards on average;

I if the Ravens run, and the Cowboys defend the pass, theRavens gain 5 yards on average;

I if the Ravens pass, and the Cowboys defend the pass, theRavens gain 0 yards on average;

I if the Ravens pass, and the Cowboys defend the run, theRavens gain 15 yards on average.

Football: Run or Pass?The Ravens are playing the Cowboys, and the Ravens are onoffense. The Ravens select a play and the Cowboys line up todefend. The choices are made simultaneously, with each teamhaving no knowledge of the other team’s choice.

Here’s a simple model: the offense chooses either to run or topass, and the defense chooses to defend a run or a defend a pass.

Based on historical data,

I if the Ravens run, and the Cowboys defend the run, theRavens lose 5 yards on average;

I if the Ravens run, and the Cowboys defend the pass, theRavens gain 5 yards on average;

I if the Ravens pass, and the Cowboys defend the pass, theRavens gain 0 yards on average;

I if the Ravens pass, and the Cowboys defend the run, theRavens gain 15 yards on average.

Football: Run or Pass?

This is a two-person (Ravens and Cowboys), zero-sumgame (what the Ravens gain, the Cowboys lose, andvice-versa). The payoff matrix, expressed from the Raven’spoint of view, is

DR DPOR −5 5OP 10 0

Here OR means that the Ravens chose to run and OPmeans they chose to pass; DR means that the Cowboyschose to defend the run and DP means they chose todefend the pass.

Constant-Sum Games

In some games, we have the same assumptions as aboveexcept that the pay-offs of both players add to a constant.

For example if both players are competing for a share of amarket of fixed size, we can write pay-offs as percentage ofthe market for each player with the percentages adding to100. All results and methods that we study for zero-sumgames also work for constant sum games.

Constant-Sum Games

In some games, we have the same assumptions as aboveexcept that the pay-offs of both players add to a constant.

For example if both players are competing for a share of amarket of fixed size, we can write pay-offs as percentage ofthe market for each player with the percentages adding to100. All results and methods that we study for zero-sumgames also work for constant sum games.

Example — percentages as payoffsRory and Corey own stores next to each other. Each daythey announce a sale giving 10% or 20% off. If they bothgive 10% off, Rory gets 70% of the customers. If Rory goesfor a 10% sale and Corey for a 20% sale, Rory gets 30% ofthe customers. If Rory announces a 20% sale and Corey a10% sale, Rory gets 90% of the customers. If they both go20%, Rory gets 50% of the customers. Between them Roryand Corey get all of the customers each day and eachcustomer patronizes only one of the shops each day.

Denote the two choices by BS = 20% off and SS = 10% off(big sale verses small sale). The payoff matrix (Rory as rowplayer) is

BS SSBS 0.5 0.9SS 0.3 0.7

Example — percentages as payoffsRory and Corey own stores next to each other. Each daythey announce a sale giving 10% or 20% off. If they bothgive 10% off, Rory gets 70% of the customers. If Rory goesfor a 10% sale and Corey for a 20% sale, Rory gets 30% ofthe customers. If Rory announces a 20% sale and Corey a10% sale, Rory gets 90% of the customers. If they both go20%, Rory gets 50% of the customers. Between them Roryand Corey get all of the customers each day and eachcustomer patronizes only one of the shops each day.

Denote the two choices by BS = 20% off and SS = 10% off(big sale verses small sale). The payoff matrix (Rory as rowplayer) is

BS SSBS 0.5 0.9SS 0.3 0.7

Example — percentages as payoffsRory and Corey own stores next to each other. Each daythey announce a sale giving 10% or 20% off. If they bothgive 10% off, Rory gets 70% of the customers. If Rory goesfor a 10% sale and Corey for a 20% sale, Rory gets 30% ofthe customers. If Rory announces a 20% sale and Corey a10% sale, Rory gets 90% of the customers. If they both go20%, Rory gets 50% of the customers. Between them Roryand Corey get all of the customers each day and eachcustomer patronizes only one of the shops each day.

Denote the two choices by BS = 20% off and SS = 10% off(big sale verses small sale). The payoff matrix (Rory as rowplayer) is

BS SSBS 0.5 0.9SS 0.3 0.7

Example — probabilities as payoffs

General Roadrunner and General Coyote are generals ofopposing armies. Every day General Roadrunner sends out abombing sortie consisting of a heavily armed bomber plane anda lighter support plane. The sortie’s mission is to drop a singlebomb on General Coyotes forces. However a fighter plane ofGeneral Coyote’s army is waiting for them in ambush and itwill dive down and attack one of the planes in the sortie.

The bomber has an 80% chance of surviving such an attack,and if it survives it is sure to drop the bomb right on the target.General Roadrunner also has the option of placing the bomb onthe support plane. In this case, due to this plane’s lighterarmament and lack of proper equipment, the bomb will reachits target with a probability of only 50% (if it is attacked byGeneral Coyote’s fighter) or 90% (if it is not). Represent thisinformation on a pay-off matrix for General Roadrunner.

Example — probabilities as payoffs

General Roadrunner and General Coyote are generals ofopposing armies. Every day General Roadrunner sends out abombing sortie consisting of a heavily armed bomber plane anda lighter support plane. The sortie’s mission is to drop a singlebomb on General Coyotes forces. However a fighter plane ofGeneral Coyote’s army is waiting for them in ambush and itwill dive down and attack one of the planes in the sortie.

The bomber has an 80% chance of surviving such an attack,and if it survives it is sure to drop the bomb right on the target.General Roadrunner also has the option of placing the bomb onthe support plane. In this case, due to this plane’s lighterarmament and lack of proper equipment, the bomb will reachits target with a probability of only 50% (if it is attacked byGeneral Coyote’s fighter) or 90% (if it is not). Represent thisinformation on a pay-off matrix for General Roadrunner.

Example — probabilities as payoffs

Coyote’s strategies are attack the bomber B or attack thesupport S. Roadrunner’s are to put the bomb on thebomber B or the support S.

If Coyote plays B and Roadrunner plays B, Coyote winswith probability 0.8. If Coyote plays B and Roadrunnerplays S, Coyote wins with probability 1.

If Coyote plays S and Roadrunner plays B, Coyote winswith probability 0.9. If Coyote plays S and Roadrunnerplays B, Coyote wins with probability .5.

Pay-off matrix:

B SB 0.8 1.0S 0.9 0.5

Example — probabilities as payoffs

Coyote’s strategies are attack the bomber B or attack thesupport S. Roadrunner’s are to put the bomb on thebomber B or the support S.

If Coyote plays B and Roadrunner plays B, Coyote winswith probability 0.8. If Coyote plays B and Roadrunnerplays S, Coyote wins with probability 1.

If Coyote plays S and Roadrunner plays B, Coyote winswith probability 0.9. If Coyote plays S and Roadrunnerplays B, Coyote wins with probability .5.

Pay-off matrix:

B SB 0.8 1.0S 0.9 0.5

Example — probabilities as payoffs

Coyote’s strategies are attack the bomber B or attack thesupport S. Roadrunner’s are to put the bomb on thebomber B or the support S.

If Coyote plays B and Roadrunner plays B, Coyote winswith probability 0.8. If Coyote plays B and Roadrunnerplays S, Coyote wins with probability 1.

If Coyote plays S and Roadrunner plays B, Coyote winswith probability 0.9. If Coyote plays S and Roadrunnerplays B, Coyote wins with probability .5.

Pay-off matrix:

B SB 0.8 1.0S 0.9 0.5

Example — probabilities as payoffs

Coyote’s strategies are attack the bomber B or attack thesupport S. Roadrunner’s are to put the bomb on thebomber B or the support S.

If Coyote plays B and Roadrunner plays B, Coyote winswith probability 0.8. If Coyote plays B and Roadrunnerplays S, Coyote wins with probability 1.

If Coyote plays S and Roadrunner plays B, Coyote winswith probability 0.9. If Coyote plays S and Roadrunnerplays B, Coyote wins with probability .5.

Pay-off matrix:

B SB 0.8 1.0S 0.9 0.5

Endgame Basketball [Ruminski]

Often in late game situations, a team with the ball mayfind themselves down by two points with the shot clockturned off. In this situation, the offensive team must decidewhether to shoot for two points, hoping to tie the game andwin in overtime, or to try for a three pointer and win thegame without overtime. The defending team must decidewhether to defend the inside or outside shot. We assumethat the probability of winning in overtime is 50% for bothteams.

In this situation, the offensive team’s coach will ask for atimeout in order to set up the play. Simultaneously, thedefensive coach will decide how to set up the defense toensure a win. Therefore we can consider this as asimultaneous move game with both coaches making theirdecisions without knowledge of the others strategy.

Endgame Basketball [Ruminski]

To calculate the probability of success for the offense,Ruminski (link to blog post) used NBA league-widestatistics on effective shooting percentages to determineprobabilities of success for open and contested shots.

Shot Success rateopen 2pt. 62.5%open 3pt. 50%

Contested 2pt. 35.7%Contested 3pt. 22.8%

Using this and the 50% probability of winning in overtimefor each team, we can figure out the probability of winningfor each team in all four scenarios using the following treediagram:

Endgame Basketball [Ruminski]

Start

Off. Shoot 2 Off. Shoot 3

Def. Defend 2 Def. Defend 3 Def. Defend 2 Def. Defend 3

Overtime

0.5 0.5

Def. wins Overtime Def. wins Def. wins Off. wins Def. wins Off. wins

Off. wins Def. wins

Start

Off. Shoot 2 Off. Shoot 3

Def. Defend 2

0.3570.643

Def. Defend 3

0.6250.375

Def. Defend 2

0.50.5

Def. Defend 3

0.2280.772

Overtime

0.5 0.5

Def. wins Overtime

0.5 0.5

Def. wins Def. wins Off. wins Def. wins Off. wins

Off. wins Def. wins Off. wins Def. wins

Endgame Basketball [Ruminski]

Start

Off. Shoot 2 Off. Shoot 3

Def. Defend 2 Def. Defend 3 Def. Defend 2 Def. Defend 3

Overtime

0.5 0.5

Def. wins Overtime Def. wins Def. wins Off. wins Def. wins Off. wins

Off. wins Def. wins

Start

Off. Shoot 2 Off. Shoot 3

Def. Defend 2

0.3570.643

Def. Defend 3

0.6250.375

Def. Defend 2

0.50.5

Def. Defend 3

0.2280.772

Overtime

0.5 0.5

Def. wins Overtime

0.5 0.5

Def. wins Def. wins Off. wins Def. wins Off. wins

Off. wins Def. wins Off. wins Def. wins

Endgame Basketball [Ruminski]

We can use those probabilities to fill in the probabilities ofa win for the row player (offense) in the payoff matrixbelow. (Note the probability for a win for the defense teamis 1 - prob. win for offense.)

Defending TeamDefend 2 Defend 3

Shoot 2Offense

Shoot 3

Endgame Basketball [Ruminski]

What are the Offense’s chances of winning if they try for 2against the inside defense? This situation starts at the left nodeof row 2. The offense wins with probability 0.357 · 0.5 = 0.1785.

What are the Offense’s chances of winning if they try for 2against the outside defense? This situation starts at the secondnode of row 2. The offense wins with probability0.625 · 0.5 = 0.3125.

What are the Offense’s chances of winning if they try for 3against the inside defense? This situation starts at the thirdnode of row 2. The offense wins with probability 0.5.

What are the Offense’s chances of winning if they try for 3against the outside defense? This situation starts at the righthand node of row 2. The offense wins with probability 0.228.

Endgame Basketball [Ruminski]

What are the Offense’s chances of winning if they try for 2against the inside defense? This situation starts at the left nodeof row 2. The offense wins with probability 0.357 · 0.5 = 0.1785.

What are the Offense’s chances of winning if they try for 2against the outside defense? This situation starts at the secondnode of row 2. The offense wins with probability0.625 · 0.5 = 0.3125.

What are the Offense’s chances of winning if they try for 3against the inside defense? This situation starts at the thirdnode of row 2. The offense wins with probability 0.5.

What are the Offense’s chances of winning if they try for 3against the outside defense? This situation starts at the righthand node of row 2. The offense wins with probability 0.228.

Endgame Basketball [Ruminski]

What are the Offense’s chances of winning if they try for 2against the inside defense? This situation starts at the left nodeof row 2. The offense wins with probability 0.357 · 0.5 = 0.1785.

What are the Offense’s chances of winning if they try for 2against the outside defense? This situation starts at the secondnode of row 2. The offense wins with probability0.625 · 0.5 = 0.3125.

What are the Offense’s chances of winning if they try for 3against the inside defense? This situation starts at the thirdnode of row 2. The offense wins with probability 0.5.

What are the Offense’s chances of winning if they try for 3against the outside defense? This situation starts at the righthand node of row 2. The offense wins with probability 0.228.

Endgame Basketball [Ruminski]

What are the Offense’s chances of winning if they try for 2against the inside defense? This situation starts at the left nodeof row 2. The offense wins with probability 0.357 · 0.5 = 0.1785.

What are the Offense’s chances of winning if they try for 2against the outside defense? This situation starts at the secondnode of row 2. The offense wins with probability0.625 · 0.5 = 0.3125.

What are the Offense’s chances of winning if they try for 3against the inside defense? This situation starts at the thirdnode of row 2. The offense wins with probability 0.5.

What are the Offense’s chances of winning if they try for 3against the outside defense? This situation starts at the righthand node of row 2. The offense wins with probability 0.228.

Endgame Basketball [Ruminski]

The pay-off matrix is

D2 D3S2 0.1785 0.3125S3 0.5 0.228

Old Exam Question

Rudolph (R) and Comet (C) play a game. They bothchoose a number between 1 and 4 simultaneously. Cometgives Rudolph a number of carrots equal to the sum of thetwo numbers chosen minus three. If this number isnegative, Comet recieves carrots from Rudolph. Which ofthe following give the pay-off matrix for Rudolph?

Old Exam Question

(a)

CNo. 1 2 3 4

1 −1 0 1 2R 2 0 1 2 3

3 1 2 3 44 2 3 4 5

(b)

CNo. 1 2 3 4

1 3 2 1 2R 2 2 1 2 3

3 1 2 3 44 2 3 4 5

(c)

CNo. 1 2 3 4

1 2 3 4 5R 2 1 0 −1 −2

3 0 1 2 34 −1 0 1 2

(d)

CNo. 1 2 3 4

1 −1 0 2 2R 2 0 0 2 0

3 1 2 3 34 2 3 4 4

(e)

CNo. 1 2 3 4

1 3 4 5 6R 2 4 5 6 7

3 5 6 7 84 6 7 8 9

Rock-Paper-Scissors in the real world

The biologists B. Sinervo and C. M. Lively wrote a reporton a lizard species whose males are divided into threeclasses according to their mating behavior. Each male ofthe side-blotched lizards (Uta Stansburiana) exhibits one ofthree (genetically transmitted) behaviors:

a) highly aggressive, with a large territory that includesseveral females;

b) monogamous, with a smaller territory that includesone female;

c) nonaggressive sneaker, with no territory, who sneaksinto the others’ domains and tries his luck with the femalesthere.

Here’s a link to a wikipedia article.

Rock-Paper-Scissors in the real world

The biologists B. Sinervo and C. M. Lively wrote a reporton a lizard species whose males are divided into threeclasses according to their mating behavior. Each male ofthe side-blotched lizards (Uta Stansburiana) exhibits one ofthree (genetically transmitted) behaviors:

a) highly aggressive, with a large territory that includesseveral females;

b) monogamous, with a smaller territory that includesone female;

c) nonaggressive sneaker, with no territory, who sneaksinto the others’ domains and tries his luck with the femalesthere.

Here’s a link to a wikipedia article.

Rock-Paper-Scissors in the real world

I the highly aggressive male has an advantage over themonogamous one (bigger, access to more resources),

I the monogamous male has an advantage over the sneaker(again, bigger & more resources), and

I the sneaker has some advantage over the highly aggressivemale, because the highly aggressive male must split histime between various consorts, and so is vulnerable tosneakers.

The observed consequence of this is that the male populationscycle from a high frequency of monogamous to a high frequencyof highly aggressives, then on to a high frequency of sneakersand back to a high frequency of monogamous.

Reference: Sinervo, B. and Lively, C. M., TheRock-Paper-Scissors Game and the evolution of alternativemale strategies, Nature, 380 (1996), 240–243. link to paper

Rock-Paper-Scissors in the real world

I the highly aggressive male has an advantage over themonogamous one (bigger, access to more resources),

I the monogamous male has an advantage over the sneaker(again, bigger & more resources), and

I the sneaker has some advantage over the highly aggressivemale, because the highly aggressive male must split histime between various consorts, and so is vulnerable tosneakers.

The observed consequence of this is that the male populationscycle from a high frequency of monogamous to a high frequencyof highly aggressives, then on to a high frequency of sneakersand back to a high frequency of monogamous.

Reference: Sinervo, B. and Lively, C. M., TheRock-Paper-Scissors Game and the evolution of alternativemale strategies, Nature, 380 (1996), 240–243. link to paper